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\begin{document}



\def\sinh{\mbox{\rm sinh}\,}
\def\cosh{\mbox{\rm cosh}\,}
\def\sh{\mbox{\rm sinh}\,}
\def\ch{\mbox{\rm cosh}\,}
\def\tan{\mbox{\rm tan}\,}
\def\tanh{\mbox{\rm tanh}\,}
\def\cotan{\mbox{\rm cotan}\,}
\def\cotanh{\mbox{\rm cotanh}\,}

\def\cF{\Cal F}
\def\cH{\Cal H}

\def\cd{\partial}  % ???


\title{A SYSTEM OF LINKED RESONATORS IN COCHLEA}

\author{S. M. Novoselova}
{Russian Academy of Sciences \\
Steklov Mathematical Institute, St.Petersburg Department\\
Fontanka 27, St.Petersburg 191011, Russia \\
E-mail: smn\@pdmi.ras.ru}



The purpose of this paper is to study  vibrations  of  a  cross
section of complex system which simulates the cochlear partition
(an element of hearing analyser of mammalian inner ear), in
terms of multy-degree-of-freedom systems.

Consider a cross section of a coupled system made up of two thin
long elastic plates linked with a raw of springs (Fig.~1).

\vskip -0.5cm\hskip -0.5cm
\special{em:graph fig1.pcx}\hspace*{8.5cm}\special{em:graph fig2.pcx}

\vspace*{4.5cm}
\begin{center}
\parbox{15cm}{\small Fig.~1. A model of cochlear complex cross section.
TM -- a plate simulating the tectorial membrane;
BM --a plate representing the basilar membrane; OHC --
the spring sketch of outer hair cell.}
\end{center}

The free vibrations of each of linked plates obey the plate equation:
\begin{equation}
D_{i}\frac{\cd^4 U_{i}}{\cd y^4}+\mu_{i}\frac{\cd^2 U_{i}}{\cd t^2}=0,
label{tag1}
\end{equation}
where $U_i(y,t)$ is normal displacement of each plate,
 $D_{i}$ is bending stiffness and  $\mu_{i}$ is surface
mass  density, i=t,b.

The gravitation force is neglected here because the original cochlea is filled
with fluid. Friction is omitted for the sake of brevity of derivations,
without loss of generality.

The long edges of the lower plate (BM) are simply supported:
\begin{equation}
U_{b}(y_0)=U_{b}(y_0+b)=0,\quad
\frac{\cd^2U_{b}(y_0)}{\cd
y^2}=\frac{\cd^2U_{b}(y_0+b)}{\cd y^2}=0.
\label{tag2}
\end{equation}

The left edge of the upper plate is also supported, while the other one
rests on the springs:

\begin{equation}
U_{t}(0)=\frac{\cd^2U_{t}(0)}{\cd y^2}=0, \qquad
Q_{t}(\eta)=D_{t}\frac{\cd^3U_{t}(\eta,t)}{\cd y^3}=\frac{F(t)}{\Delta x},
\qquad M_{t}(\eta)=Q_{t}(\eta)\cdot\eta.
\label{tag3}
\end{equation}
Here $Q_t$ and $M_t$ are the crosscutting force and bending moment, ${\Delta x}$
is the distance between two adjacent springs in longitudinal direction,
$F(t)$ is the force acting onto the spring.

The force $F$ is
proportional to variation of distance between the plates:
\begin{equation}
F(t)=S(U_{t}-U_{b})\big|_{y=\eta}+m\frac{\cd^2(U_{t}-U_{b})}
 {\cd t^2}\bigg|_{y=\eta}
\label{tag4}
\end{equation}
Here $m$ denotes mass of the spring and $S$ is the restoring spring
constant.

The force $F$ also affects the lower plate causing a break
of crosscutting force under the line of springs:
\begin{equation}
Q_{b}(\eta+0)-Q_{b}(\eta-0)=-\frac F{\Delta x}.
\label{tag5}
\end{equation}

Solution to the problem (1)-(5) is obtained with use of Fourier method.
For the sake of simplicity and brevity of derivations and without
loss of generality we assume:
$$
\eta=b, \qquad
y_0=\eta/2.
$$
The eigenfunctions of each plate are supposed to be sums of
trigonometrical and hyperbolical sines and cosines. The boundary and
continuity conditions define initially indeterminate coefficients:
$$
W_t=a_{1t}\left(\sin k_{t}y+\frac{\sin\lambda_t-\lambda_{t}\cos\lambda_{t}}
{\sinh\lambda_{t}-\lambda_{t}\cosh\lambda_{t}} \,\sinh k_{t}y\right),
$$
\begin{equation}
W_{b}=a_{1b}\left\{
\begin{array}{l l}
\sin k_b(y-y_0)-\frac{\cos\lambda_b}{\cosh\lambda_b}\,\sinh k_b(y-y_0)\
&\mbox{if }\ y_0\le y\le \eta\\
\sin k_b(y_0+b-y)-\frac{\cos\lambda_b}{\cosh\lambda_b}\,\sin k_{b}(y_0+b-y)\
&\mbox{if }\ \eta\le y\le y_0+b
\end{array}\right.
\label{tag6}
\end{equation}
where
\begin{equation}
\lambda_t=k_t\eta,\quad \lambda_b=k_bb/2,\quad k_i^4=\omega_0^2\mu_i/D_i
\label{tag7}
\end{equation}

Eigenfrequency $\omega_0$ of complex cross section is found to be split:
\begin{equation}
\omega_0^2=\omega_c^2\pm\frac{\Delta x/m}{r_b\pm r_t}
\label{tag8}
\end{equation}
where:
\begin{equation}
\omega_c^2=S/m,\quad
r_{b}=\frac{\tan\lambda_{b}-\tanh\lambda_{b}}{4D_{b}k^3_{b}},\quad
r_{t}=\frac{2-\lambda_{t}(\cotan\lambda_{t}+\cotanh\lambda_{t})}
{D_{t}k^3_{t}(\cotan\lambda_{t}-\cotanh\lambda_{t})}.
\label{tag9}
\end{equation}

The sign $\pm$ depends on the initial shake,  which  can  provide
synphase (-)
or counterphase (+) vibrations of two plates. Effect of splitting
eigenfrequency is specific to the multi-degree-of-freedom systems.

Suppose the cross section be a subject to a periodic force
$P_0e^{i\omega t}$ constant in $Y$
direction and applied, for example, from below:
\begin{equation}
D_{b}\frac{\cd^4U_{b}}{\cd y^4}-\mu_{b}\omega^2U_{b}=P_0e^{i\omega t},\quad
D_{t}\frac{\cd^4 U_{t}}{\cd y^4}-\mu_{t}\omega^2U_{t}=0.
\label{tag10}
\end{equation}

It is easy to verify that the eigenfunctions $W_{b}$ and $W_{t}$ do not
form a set of orthogonal functions, so the traditional method of expansion
of right-hand part
into the eigenfunction series is not applicable here.
However, in close vicinities of $\lambda_b=\pm(\pi/2+m\pi),\ m=0,1,2...$ the
eigenfunctions $W_{bm}$ are very close to sines. In these narrow region,
the approximate equality holds:
\begin{equation}
U_b\approx\sum_{m=0}^{\infty}c_{bm}\sin k_{bm}(y-y_0)\exp(i\omega t), \qquad
k_{bm}=\pm(\pi+2m\pi)/b,
\label{tag11}
\end{equation}
where $c_{bm}$ are the indeterminate coefficients.
The wavenumbers (11) are also the eigenvalues of free vibrations
of uncoupled plate BM.

Substituting of (11) into (10) and using (7) and (8) yields:
\begin{equation}
c_{bm}=\frac{2P_0}
{\mu_b\lambda_{bm}(\omega_c^2\pm\frac{\Delta x/m}{r_{bm}\pm r_{tm}}-\omega^2)}
\label{tag12}
\end{equation}
The second term of the last factor of
denominator of (12) tends to zero at $\lambda_b=\pm(\pi/2+m\pi)$. The
amplitude coefficient $c_m$ has a maximum if $\omega=\omega_c$, otherwise, when
the resonances of  isolated  spring  and  uncoupled  lower  plate
coincide.
Similar relationship can be derived for the upper plate in vicinity of its
partial eigenfrequency, using the series of
orthogonal eigenfunctions of the plate
with free right edge.

The more general case, when $\omega=\omega_0\ne\omega_c$, can not be examined
in the above determined narrow regions, but such a case, if
exists, should not entail any principally different results.

Such approach allows to clarify a great deal of
experimental phenomena in hearing by simple and natural way [1]. In
particular, Eq.(8) allows to explain a paradoxical experimental findings
that the eigenfrequency of cochlear cross section falls under
overstimulating, while the stiffness of isolated outer hair cell (modelled
with a spring here) does increase [2]. Such unusual behavior served as
argument for lots of negative dumping models. However, a key to the
problem could lie in the fact, that the sound pressure is applied to the
cochlear partition in such a way, that it produced only the synphase
vibrations of two plates. When stiffness $S$, and accordingly, $\omega_c$
 increase
under the high energy level, the second term of (8) increases in turn, and
with negative sign. Under certain relationship between parameters of the
plate and spring, the second term may increase more fast, thus reducing
the sum. Moreover, as usual, in multi-degree-of-freedom systems with equal
partial eigenfrequencies of subsystems, the synphase eigenfrequency is
lower and the counterphase one is higher than the partial eigenfrequency[3].

Supported by Russian Basic Research Foundation, Grant 93-011-16148.

\vskip -0.5cm\begin{thebibliography}{99}
\vskip -0.2cm
\bibitem{1.} Novoselova,~S.~M. (1994) ``The transversal modes of
tectorial membrane free vibrations and origin of the outer hair
cells oscillations.'' {\it Zap. Nauchn. Semin. POMI}, {\bf 218}, pp 138--148.
English translation has to appear in J. of Mathematical Sciences.
\vskip -0.2cm
\bibitem{2.}
De Boer,~E. (1991)
" Auditory physics. Physical principles in
hearing theory. III ",
{\it Physics reports} {\bf 203(3)} pp 125--231
\vskip -0.2cm
\bibitem{3.}
Haikin,~C.~E. (1971) "The physical grounds of Mechanics", Moscow,
"Science", in Russian.
\end{thebibliography}
\end{document}
\end{document}
